In this paper we construct the Floer homology for an action functional which was introduced by Rabinowitz and prove a vanishing theorem. As an application, we show that there are no displaceable exact contact embeddings of the unit cotangent bundle of a sphere of dimension greater than three into a convex exact symplectic manifold with vanishing first Chern class. This generalizes Gromov's result that there are no exact Lagrangian embeddings of a sphere into ރ n .
Abstract. The first two authors have recently defined RabinowitzFloer homology groups RF H * (M, W ) associated to an exact embedding of a contact manifold (M, ξ) into a symplectic manifold (W, ω). These depend only on the bounded component V of W \ M . We construct a long exact sequence in which symplectic cohomology of V maps to symplectic homology of V , which in turn maps to Rabinowitz-Floer homology RF H * (M, W ), which then maps to symplectic cohomology of V . We compute RF H * (ST * L, T * L), where ST * L is the unit cosphere bundle of a closed manifold L. As an application, we prove that the image of an exact contact embedding of ST * L (endowed with the standard contact structure) cannot be displaced away from itself by a Hamiltonian isotopy, provided dim L ≥ 4 and the embedding induces an injection on π1. In particular, ST * L does not admit an exact contact embedding into a subcritical Stein manifold if L is simply connected. We also prove that Weinstein's conjecture holds in symplectic manifolds which admit exact displaceable codimension 0 embeddings. IntroductionLet (W, λ) be a complete convex exact symplectic manifold, with symplectic form ω = dλ (see Section 3 for the precise definition). An embedding ι : M ֒→ W of a contact manifold (M, ξ) is called exact contact embedding if there exists a 1-form α on M such that such that ker α = ξ and α − λ| M is exact. We identify M with its image ι(M ). We assume that W \ M consists of two connected components and denote the bounded component of W \ M by V . One can classically [25] associate to such an exact contact embedding the symplectic (co)homology groups SH * (V ) and SH * (V ). We The first two authors have recently defined for such an exact contact embedding Floer homology groups RF H * (M, W ) for the Rabinowitz action functional [9]. We refer to Section 3 for a recap of the definition and of some useful properties. We will show in particular that these groups do not depend on W , but only on V (the same holds for SH * (V ) and SH * (V )). We shall use in this paper the notation RF H * (V ) and call them Rabinowitz Floer homology groups. Remark 1.1. All (co)homology groups are taken with field coefficients. Without any further hypotheses on the first Chern class c 1 (V ) of the tangent bundle, the symplectic (co)homology and Rabinowitz Floer homology groups are Z 2 -graded. If c 1 (V ) = 0 they are Z-graded, and if c 1 (V ) vanishes on π 2 (V ) the part constructed from contractible loops is Z-graded. This Zgrading on Rabinowitz Floer homology differs from the one in [9] (which takes values in 1 2 + Z) by a shift of 1/2 (see Remark 3.2).Our purpose is to relate these two constructions. The relevant object is a new version of symplectic homology, denotedŠH * (V ), associated to " -shaped" Hamiltonians like the one in Figure 1 on page 20 below. This version of symplectic homology is related to the usual ones via the long exact sequence in the next theorem. Theorem 1.2. There is a long exact sequence (1). . .The long exact sequence (1) can be seen ...
We study the dynamics and symplectic topology of energy hypersurfaces of mechanical Hamiltonians on twisted cotangent bundles. We pay particular attention to periodic orbits, displaceability, stability and the contact type property, and the changes that occur at the Mañé critical value c . Our main tool is Rabinowitz Floer homology. We show that it is defined for hypersurfaces that are either stable tame or virtually contact, and that it is invariant under homotopies in these classes. If the configuration space admits a metric of negative curvature, then Rabinowitz Floer homology does not vanish for energy levels k > c and, as a consequence, these level sets are not displaceable. We provide a large class of examples in which Rabinowitz Floer homology is nonzero for energy levels k > c but vanishes for k < c , so levels above and below c cannot be connected by a stable tame homotopy. Moreover, we show that for strictly 1=4-pinched negative curvature and nonexact magnetic fields all sufficiently high energy levels are nonstable, provided that the dimension of the base manifold is even and different from two. 53D40; 37D40
We show that the planar circular restricted three-body problem is of restricted contact type for all energies below the first critical value (action of the first Lagrange point) and for energies slightly above it. This opens up the possibility of using the technology of contact topology to understand this particular dynamical system.
In this article we explain how critical points of a particular perturbation of the Rabinowitz action functional give rise to leaf-wise intersection points in hypersurfaces of restricted contact type. This is used to derive existence and multiplicity results for leaf-wise intersection points in hypersurfaces of restricted contact type in general exact symplectic manifolds. The notion of leaf-wise intersection points was introduced by Moser [Mos78].2000 Mathematics Subject Classification. 53D40, 37J10, 58J05. Key words and phrases. Leaf-wise Intersections, Rabinowitz Floer homology, stretching the neck, local homology. A perturbation of the Rabinowitz action functionalWe recall that Σ ⊂ (M, ω = dλ) is a closed hypersurface in an exact symplectic manifold such that (Σ, α = λ |Σ ) is a contact manifold. Moreover, Σ is assumed to bound a compact region in M . We denote by R the Reeb vector field of α. Moreover, we define the vector field Y by dλ(Y, ·) = λ(·).Lemma 2.1. The vector field Y is a Liouville vector field for (Σ, α), that is, L Y ω = ω and Y ⋔ Σ. In particular, (Σ, α) is of restricted contact type.
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